Monotonic function

In mathematics, a function (mathematics) is monotonic or monotone increasing if preserves order: that is, if inputs x and y satisfy $x\leq y$ then the outputs from f satisfy $f(x)\leq f(y)$. A monotonic decreasing function similarly reverses the order. A function is strictly monotonic if inputs x and y satisfying $x<y$ have outputs from f satisfying $f(x)<f(y)$: that is, it is injective in addition to being montonic.

A special case of a monotonic function is a sequence regarded as a function defined on the natural numbers. So a sequence $a_{n}$ is monotonic increasing if $m\leq n$ implies $a_{m}\leq a_{n}$. In the case of real sequences, a monotonic sequence converges if it is bounded. Every real sequence has a monotonic subsequence.

A differentiable function on the real numbers is monotonic when its derivative is non-zero: this is a consequence of the Mean Value Theorem.